Questions
The solution of the differential equation
$\frac{dy}{dx}$+ 2xy = $e^{-x^2}$
- (1 + x) $e^{+x^2}$
- (1 + x) $e^{-x^2}$
- (1 - x) $e^{-x^2}$
- (1 - x) $e^{+x^2}$
For a four-bar linkage in toggle position, the value of mechanical advantage is
- 0.0
- 0.5
- 1.0
- $\infty$
Let x denote a real number. Find out the INCORRECT statement from the following, where a and b are real numbers.
- S = {x : x > 3} represents the set of all real numbers greater than 3.
- S = {x : x2 < 0} represents the empty set.
- S = {x : x $\in$A and x $\in$ B} represents the union of set A and set B.
- S = {x : a < x < b} represents the set of all real numbers between a and b.
Match the items in columns I and II.<blockquote |||="" |---|---|="" |="" <b="">Column I| Column II|
| (P) Gauss-Seidel method| (1) Interpolation|
| (Q) Forward Newton-Gauss method| (2) Non-linear differential equations|
| (R) Runge-Kutta method| (3) Numerical integration|
| (S) Trapezoidal Rule| (4) Linear algebraic equations|
- P - 1, Q - 4, R - 3, S - 2
- P - 1, Q - 4, R - 2, S - 3
- P - 1, Q - 3, R - 2, S - 4
- P - 4, Q - 1, R - 2, S - 3
A pin-ended column of length L, modulus of elasticity E and second moment of the cross-sectional area I is loaded centrically by a compressive load P. The critical buckling load $|(P_{cr})|$ is given by
- $P_{cr}=\frac{EI}{x^2L^2}$
- $P_{cr}=\frac{x^2EI}{3L^2}$
- $P_{cr}=\frac{\pi EI}{L^2}$
- $P_{cr}=\frac{x^2EI}{L^2}$
For a Newtonian fluid,
- shear stress is proportional to shear strain
- rate of shear stress is proportional to shear strain
- shear stress is proportional to rate of shear strain
- rate of shear stress is proportional to rate of shear strain
Equation of the line normal to function
f (x) = (x − 8)2/3 + 1 at P (0, 5) is
- y = 3x − 5
- y = 3x + 5
- 3y = x + 15
- 3y = x − 15
In an MRP system, component demand is
- forecasted
- established by the master production schedule
- calculated by the MRP system from the master production schedule
- ignored
Match the items in columns I and II.
- P - 3, Q - 1, R - 4, S - 2
- P - 2, Q - 3, R - 4, S - 1
- P - 3, Q - 2, R - 5, S - 4
- P - 3, Q - 4, R - 2, S - 1
Eigen values of a matrix S = $\begin{bmatrix}
\ 3 & 2 \
\ 2 & 3 \
\end{bmatrix}$are 5 and 1. What are the eigen values of the matrix S2 = SS?
- 1 and 25
- 6 and 4
- 5 and 1
- 2 and 10
A disk clutch is required to transmit 5 kW at 2000 rpm. The disk has a friction lining with coefficient of friction equal to 0.25, bore radius of friction lining is equal to 25 mm. Assume uniform contact pressure of 1 MPa. The value of outside radius of the friction lining is
- 39.4 mm
- 49.5 mm
- 97.9 mm
- 142.9 mm
In a composite slab, the temperature at the interface (Tinter) between two materials is equal to the average of the temperatures at the two ends. Assuming steady one-dimensional heat conduction, which of the following statements is true about the respective thermal conductivities?
- 2k1 = k2
- k1 = k2
- 2k1 = 3k2
- k1 = 2k2
Match the items in columns I and II.
- P - 4, Q - 2, R - 3, S - 1
- P - 4, Q - 3, R - 2, S - 1
- P - 3, Q - 2, R - 1, S - 4
- P - 3, Q - 4, R - 1, S - 2
Twenty degree full depth involute profiled 19-tooth pinion and 37-tooth gear are in mesh. If the module is 5 mm, the center distance between the gear pair will be
- 140 mm
- 150 mm
- 280 mm
- 300 mm
A steel bar of 40 mm × 40 mm square cross-section is subjected to an axial compressive load of 200 kN. If the length of the bar is 2 m and E = 200 GPa, then the elongation of the bar will be
- 1.25 mm
- 2.70 mm
- 4.05 mm
- 5.40 mm
For $\frac{d^2y}{dx^2}+4\frac{dy}{dx}$+ 3y = 3e2x, the particular integral is
- $\frac{1}{15}e^{2X}$
- $\frac{1}{15}e^{2X}$
- 3e2x
- C1 e-x + C1 e-3x
The differential equation governing the vibrating system is
- m$\ddot X $ + c$\ddot X $ + k (x − y) = 0
- m ($\ddot X $ - $\ddot y $) + c ($\ddot X $ - $\dot y $) + kx = 0
- m $\ddot X $ + c ($\dot X $ - $\dot y $) + kx = 0
- m ($\ddot X $ - $\ddot y $) + c ($\dot X $ - $\dot y $) + k (x - y) = 0
According to Von-Mises' distortion energy theory, the distortion energy under three dimensional stress state is represented by
- $\frac{1}{2E}[\sigma_1^2+\sigma_2^2+\sigma_2^2-2V(\sigma_1\sigma_2+\sigma_3\sigma_2+\sigma_1\sigma_3)]$
- $\frac{1+v}{2E}[\sigma_1^2+\sigma_2^2+\sigma_2^2-(\sigma_1\sigma_2+\sigma_3\sigma_2+\sigma_1\sigma_3)]$
- $\frac{1+v}{3E}[\sigma_1^2+\sigma_2^2+\sigma_2^2-(\sigma_1\sigma_2+\sigma_3\sigma_2+\sigma_1\sigma_3)]$
- $\frac{1}{3E}[\sigma_1^2+\sigma_2^2+\sigma_2^2-V(\sigma_1\sigma_2+\sigma_3\sigma_2+\sigma_1\sigma_3)]$
The number of customers arriving at a railway reservation counter is Poisson distributed with an arrival rate of eight customers per hour. The reservation clerk at this counter takes six minutes per customer on an average with an exponentially distributed service time. The average number of the customers in the queue will be
- 3
- 3.2
- 4
- 4.2
A machine of 250 kg mass is supported on springs of total stiffness 100 kN/m. Machine has an unbalanced rotating force of 350 N at speed of 3600 rpm. Assuming a damping factor of 0.15, the value of transmissibility ratio is
- 0.0531
- 0.9922
- 0.0162
- 0.0028
In a Pelton wheel, the bucket peripheral speed is 10 m/s, the water jet velocity is 25 m/s and volumetric flow rate of the jet is 0.1 m3/s. If the jet deflection angle is 120° and the flow is ideal, the power developed is
- 7.5 kW
- 15.0 kW
- 22.5 kW
- 37.5 kW
A bar having a cross-sectional area of 700 mm2 is subjected to axial loads at the positions indicated. The value of stress in the segment QR is
- 40 MPa
- 50 MPa
- 70 MPa
- 120 MPa
A large hydraulic turbine is to generate 300 kW at 1000 rpm under a head of 40 m. For initial testing, a 1:4 scale model of the turbine operates under a head of 10 m. The power generated by the model (in kW) will be
- 2.34
- 4.68
- 9.38
- 18.75
If a system is in equilibrium and the position of the system depends upon many independent variables, the principle of virtual work states that the partial derivatives of its total potential energy with respect to each of the independent variable must be
- -1.0
- 0
- 1.0
- $\infty$
The velocity profile in fully developed laminar flow in a pipe of diameter D is given by u = u0 (1 - 4r2/ D2), where r is the radial distance from the center. If the viscosity of the fluid is F, the pressure drop across a length L of the pipe is
- $\frac{\mu u_0L}{D^2}$
- $\frac{4\mu u_0L}{D^2}$
- $\frac{8\mu u_0L}{D^2}$
- $\frac{16\mu u_0L}{D^2}$
A two-dimensional flow field has velocities along the x and y directions given by u = x2 t and v = -2xyt respectively, where t is time. The equation of streamlines is
- x2 y = constant
- xy2 = constant
- xy = constant
- Not possible to determine
A siphon draws water from a reservoir and discharges it out at atmospheric pressure. Assuming ideal fluid and the reservoir is large, the velocity at point P in the siphon tube is
- $\sqrt{2gh_1}$
- $\sqrt{2gh_2}$
- $\sqrt{2g(h_2+h_1)}$
- $\sqrt{2g(h_2+h_1)}$
In a four-bar linkage, S denotes the shortest link length, L is the longest link length, P and Q are the lengths of other two links. At least one of the three moving links will rotate by 360° if
- S + L $\leq$ P + Q
- S + L > P + Q
- S + P $\leq$ L + Q
- S + P > L + Q
If point A is in equilibrium under the action of the applied forces, the values of tensions and T AB, T AC are respectively.
- 520 N and 300 N
- 300 N and 520 N
- 450 N and 150 N
- 150 N and 450 N
A 100 W electric bulb was switched on in a 2.5 m × 3 m × 3 m size thermally insulated room having a temperature of 20°C. The room temperature at the end of 24 hours will be
- 321 °C
- 341 °C
- 450 °C
- 470 °C
Given below is an extract from steam tables.
Specific enthalpy of water in kJ/kg at 150 bar and 45 °C is
- 203.60
- 200.53
- 196.38
- 188.45
Match the items in columns I and II.
| Column - I | Column - II |
| P. Higher Kinematic Pair | 1. Grubler's Equation |
| Q. Lower Kinemation Pair | 2. Line contact |
| R. Quick Return Mechanism | 3. Euler's Equation |
| S. Mobility of a Linkage | 4. Planar |
| 5. Shaper | |
| 6. Surface contact |
- P - 2, Q - 6, R - 4, S - 3
- P - 6, Q - 2, R - 4, S - 1
- P - 6, Q - 2, R - 5, S - 3
- P - 2, Q - 6, R - 5, S - 1
Determine the correctness or otherwise of the following Assertion [a] and the Reason [r].
Assertion [a]: In a power plant working on a Ranking cycle, the regenerative feed water heating improves the efficiency of the steam turbine.
Reason [r]: The regenerative feed water heating raises the average temperature of heat addition in the Rankine cycle.
- Both [a] and [r] are true and [r] is the correct reason for [a].
- Both [a] and [r] are true but [r] is NOT the correct reason for [a].
- Both [a] and [r] are false
- [a] is false and [r] is true.
Determine the correctness or otherwise of the following Assertion [a] and the Reason [r].
Assertion [a]: Condenser is an essential equipment in a steam power plant.
Reason [r]: For the same mass flow rate and the same pressure rise, a water pump requires substantially less power than a steam compressor.
- Both [a] and [r] are true and [r] is the correct reason for [a].
- Both [a] and [r] are true but [r] is NOT the correct reason for [a].
- [a] is true but [r] is false
- [a] is true but [r] is false
A horizontal shaft centrifugal pump lifts ware at 65°C. The suction nozzle is one meter below pump centerline. The pressure at this point equal 200 kPa gauge and velocity is 3 m/s. Steam tables show saturation pressure at 65°C is 25 kPa, and specific volume of the saturated liquid is 0.001020 m3/kg. The pump Net Positive suction Head (NPSH) is
- 24 m
- 26 m
- 28 m
- 30 m
Match items from groups I, II, III, IV and V.
- F - G - J - K - M ; E - G - I - K - N
- E - G - I - K - M ; F - H - I - K - N
- F - H - J - L - N ; E - H - I - L - M
- E - G - J - K - N ; F - H - J - K - M
Group I shows different heat addition processes in power cycles. Likewise, Group II shows different heat removal processes. Group III lists power cycles. Match items from Groups I, II and III.
- P - S - 5, R - U - 3, P - S - 1, Q - T - 2
- P - S - 1, R - U - 3, P - S - 4, P - T - 2
- R - T - 3, P - S - 1, P - T - 4, Q - S - 5
- P - T - 4, R - S - 3, P - S - 1, P - S - 5
With an increase in thickness of insulation around a circular pipe, heat loss to surroundings due to
- convection increases, while that due to conduction decreases
- convection decreases, while that due to conduction increases
- convection and conduction decreases
- convection and conduction increases
Consider a continuous random variable with probability density function as follows.
f (t) = 1 + t for - 1 $\leq$ t $\leq$ 0
= 1 - t for 0 $\leq$ t $\leq$ 1
The standard deviation of the random variable is
- $\frac{1}{\sqrt{3}}$
- $\frac{1}{\sqrt{5}}$
- $\frac{1}{3}$
- $\frac{1}{6}$
In a wire drawing operation, diameter of a steel wire is reduced from 10 mm to 8 mm. The mean flow stress of the material is 400 MPa. The ideal force required for drawing (ignoring friction and redundant work) is
- 4.48 kN
- 8.97 kN
- 20.11 kN
- 31.41 kN
Match the items in columns I and II.
- P - 6, Q - 3, R - 1, S - 2
- P - 4, Q - 5, R - 6, S - 1
- P - 2, Q - 5, R - 3, S - 4
- P - 4, Q - 3, R - 1, S - 2
The table gives details of an assembly line.
What is the line efficiency of the assembly line?
- 70%
- 75%
- 80%
- 85%
A stockist wishes to optimize the number of perishable items he needs to stock in any month in his store. The demand distribution for this perishable item is given in the table:
The stockist pays Rs. 70 for each item and he sells each at Rs. 90. If the stock is left unsold in any month, he can sell the item at Rs. 50 each. There is no penalty for unfulfilled demand. To maximize the expected profit, the optimal stock level is
- 5 units
- 4 units
- 3 units
- 2 units
A thin layer of water in field is formed after a farmer has watered it. The ambient air conditions are: temperature 20°C and relative humidity 5%. An extract of steam tables is given below
| Temperature (<span style="font-size:" 11px;="">oC) | - 15 | - 10 | - 5 | 0.01 | 5 | 10 | 15 | 20 |
| Saturation pressure (kPa) | 0.10 | 0.26 | 0.40 | 0.61 | 0.87 | 1.23 | 1.71 | 2.34 |
Neglecting the heat transfer between the water and the ground, the water temperature in the field after phase equilibrium is reached equals
- 10.3°C
- -10.3°C
- -14.5°C
- 14.5°C
Consider the following data for an item:
Annual demand: 2500 units per year
Ordering cost: Rs.100 per order
Inventory holding rate: 25% of unit price
The optimum order quantity (in units) is
- 447
- 471
- 500
- $\geq$600
A manufacturing shop processes sheet metal jobs, wherein each job must pass through two machines (M1 and M2, in that order). The processing time (in hours) for these jobs is give in the table:
The optimal make-span (in hours) of the shop is
- 120
- 115
- 109
- 79
A simply supported beam of span length 6 m and 75 mm diameter carries a uniformly distributed load of 1.5 kN/m.
What is the maximum value of bending moment?
- 6.75 kNm
- 13.5 kNm
- 81 kNm
- 125 kNm
A simply supported beam of span length 6 m and 75 mm diameter carries a uniformly distributed load of 1.5 kN/m.
What is the maximum value of bending stress?
- 162.98 MPa
- 325.95 MPa
- 625.95 MPa
- 651.90 MPa
A vibratory system consists of a mass 12.5 kg, a spring of stiffness 1000 N/m, and a dashpot with damping coefficient of 15 Ns/m.
The value of critical damping of the system is
- 0.223 Ns/m
- 17.88 Ns/m
- 71.4 Ns/m
- 223.6 Ns/m
A vibratory system consists of a mass 12.5 kg, a spring of stiffness 1000 N/m, and a dashpot with damping coefficient of 15 Ns/m.
The value of logarithmic decrement is
- .422
- 1.32
- 0.68
- 0.66
A planetary gear train has four gears and one carrier. Angular velocities of the gears are $\infty_1$,$\infty_2$,$\infty_3$ and$\infty_4$ respectively. The carrier rotates with angular velocity $\infty_5$.
For $\infty_1$= 60 rpm clockwise (cw) when looked from the left, what is the angular velocity of the carrier and its direction so that Gear 4 rotates in counterclockwise (ccw) direction at twice the angular velocity of Gear 1 when looked from the left?
- 130 rpm, cw
- 223 prm, ccw
- 256 rpm, cw
- 156 rpm, ccw
A firm is required to procure three items (P, Q and R). The prices quoted for these items (in Rs.) by suppliers S1, S2 and S3 are given in table. The management policy requires that each item has to be supplied by only one supplier and one supplier supply only one item. The minimum total cost (in Rs.) of procurement to the firm is
- 350
- 360
- 385
- 395
A football was inflated to a gauge pressure of 1 bar when the ambient temperature was 15°C. When the game started next day, the air temperature at the stadium was 5°C. Assume that the volume of the football remains constant at 2500 cm3.
Gauge pressure of air to which the ball must have been originally inflated so that it would equal 1 bar gauge at the stadium is
- 2.23 bar
- 1.94 bar
- 1.07 bar
- 1.00 bar
A football was inflated to a gauge pressure of 1 bar when the ambient temperature was 15°C. When the game started next day, the air temperature at the stadium was 5°C. Assume that the volume of the football remains constant at 2500 cm3.
The amount of heat lost by the air in the football and the gauge pressure of air in the football at the stadium are respectively equal to
- 30.6 J, 1.94 bar
- 21.8 J, 0.93 bar
- 61.1 J, 1.94 bar
- 43.7 J, 0.93 bar
Consider a PERT network for a project involving six tasks (a to f).
The expected completion time of the project is
- 238 days
- 224 days
- 171 days
- 155 days
A smooth flat plate with a sharp leading edge is placed along a gas stream flowing at U = 10 m/s. The thickness of the boundary layer at section r − s is 10 mm, the breadth of the plate is 1 m (into the paper) and the density of the gas $\rho$= 1.0 kg/m. Assume that the boundary layer is thin, two-dimensional and follows a linear velocity distribution, u = U (y/$\delta$) , at the section r − s , where y is the height from plate.
The integrated drag force (in N) on the plate between p − s is:
- 0.67
- 0.33
- 0.17
- zero
A smooth flat plate with a sharp leading edge is placed along a gas stream flowing at U = 10 m/s. The thickness of the boundary layer at section r − s is 10 mm, the breadth of the plate is 1 m (into the paper) and the density of the gas $\rho$= 1.0 kg/m. Assume that the boundary layer is thin, two-dimensional, and follows a linear velocity distribution, u = U (y/$\delta$) , at the section r − s , where y is the height from plate.
The mass flow rate (in kg/s) across the section q − r is
- zero
- 0.05
- 0.10
- 0.15
Consider a PERT network for a project involving six tasks (a to f).
The standard deviation of the critical path of the project is
- $\sqrt{151}$ days
- $\sqrt{155}$ days
- $\sqrt{200}$ days
- $\sqrt{238}$ days
A ring gage is used to measure
- outside diameter but not roundness
- roundness but not outside diameter
- both outside diameter and roundness
- only external threads
NC contouring is an example of
- continuous path positioning
- point-to-point positioning
- absolute positioning
- incremental positioning
An expendable pattern is used in
- slush casting
- squeeze casting
- centrifugal casting
- investment casting
The number of inversions for a slider crank mechanism is
- 6
- 5
- 4
- 3
The ultimate tensile strength of a material is 400 MPa and the elongation up to maximum load is 35%. If the material obeys power law of hardening, then the true stress-true strain relation (stress in MPa) in the plastic deformation range is
- $\sigma=540\in ^{0.30}$
- $\sigma=775\in ^{0.30}$
- $\sigma=540\in ^{0.35}$
- $\sigma=775\in^{0.35}$
A 4 mm thick sheet is rolled with 300 mm diameter rolls to reduce thickness without any change in its width. The friction coefficient at the work-roll interface is 0.1. The minimum possible thickness of the sheet that can be produced in a single pass is
- 1.0 mm
- 1.5 mm
- 2.5 mm
- 3.7 mm
In a sand casting operation, the total liquid head is maintained constant such that it is equal to the mould height. The time taken to fill the mould with a top gate is tA. If the same mould is filed with a bottom gate, then the time taken is tB. Ignore the time required to fill the runner and frictional effects. Assume atmospheric pressure at the top molten metal surfaces. The relation between and tA tB is
- tB = $\sqrt{2}$tA
- tB = 2 tA
- tB = $\frac{t_A}{\sqrt{2}}$tA
- tB = 2$\sqrt{2}$tA
The main purpose of spheroidising treatment is to improve
- hardenability of low carbon steels
- machinability of low carbon steels
- hardenability of high carbon steels
- machinability of high carbon steels
In an arc welding process, the voltage and current are 25 V and 300 A respectively. The arc heat transfer efficiency is 0.85 and welding speed is 8 mm/sec. The net heat input (in J/mm) is
- 64
- 797
- 1103
- 79700
If each abrasive grain is viewed as a cutting tool, which of the following represents the cutting parameters in common grinding operations?
- Large negative rake angle, low shear angle and high cutting speed
- Large positive rake angle, low shear angle and high cutting speed
- Large negative rake angle, high shear angle and low cutting speed
- Zero rake angle, high shear angle and high cutting speed
Arrange the processes in the increasing order of their maximum material removal rate.
Electrochemical Machining (ECM)
Ultrasonic Machining (USM)
Electron Beam Machining (EBM)
Laser Beam Machining (LBM) and
Electric Discharge Machining (EDM)
- USM, LBM, EBM, EDM, ECM
- EBM, LBM, USM, ECM, EDM
- LBM, EBM, USM, ECM, EDM
- LBM, EBM, USM, EDM, ECM
Match the items in columns I and II.
- P - 4, Q - 5, R - 3, S - 2
- P - 3, Q - 5, R - 1, S - 4
- P - 2, Q - 4, R - 3, S - 5
- P - 4, Q - 2, R - 1, S - 3
Uncut thickness = 0.5 mm, Cutting speed = 20 m/min, Width of cut = 5 mm, Chip thickness = 0.7 mm
Thrust force = 200 N, Cutting force = 1200 N, Rake angle = 15°
Assume Merchant's theory.
The values of shear angle and shear strain, respectively, are
- 30.3° and 1.98
- 30.3° and 4.23
- 40.2° and 2.97
- 40.2° and 1.65
Uncut thickness = 0.5 mm, Cutting speed = 20 m/min, Width of cut = 5 mm, Chip thickness = 0.7 mm
Thrust force = 200 N, Cutting force = 1200 N, Rake angle = 15�
Assume Merchant's theory.
The coefficient of friction at the tool-chip interface is
- 0.23
- 0.46
- 0.85
- 0.95
Uncut thickness = 0.5 mm, Cutting speed = 20 m/min, Width of cut = 5 mm, Chip thickness = 0.7 mm
Thrust force = 200 N, Cutting force = 1200 N, Rake angle = 15�
Assume Merchant's theory.
The percentage of total energy dissipated due to friction at the tool-chip interface is
- 30%
- 42%
- 58%
- 70%
Dew point temperature is the temperature at which condensation begins when the air is cooled at constant.
- volume
- entropy
- pressure
- enthalpy
The statement concern psychrometric chart.
- Constant relative humidity lines are uphill straight lines to the right.
- Constant wet bulb temperature lines are downhill straight lines to the right.
- Constant specific volume lines are downhill straight lines to the right.
- Constant enthalpy lines are coincident with constant wet bulb temperature lines.
Which of the statements are correct?
- 2 and 3
- 1 and 2
- 1 and 3
- 2 and 4
A cylindrical shaft is subjected to an alternating stress of 100 MPa. Fatigue strength to sustain 1000 cycle is 490 MPa. If the corrected endurance strength is 70 MPa, estimated shaft life will be
- 1071 cycles
- 15000 cycles
- 281914 cycles
- 928643 cycles
A 60 mm long and 6 mm thick fillet weld carries a steady load of 15 kN along the weld. The shear strength of the weld material is equal to 200 MPa. The factor of safety is
- 2.4
- 3.4
- 4.8
- 6.8
If f Cf is the coefficient of speed fluctuation of a flywheel then the ratio of $\omega_{max}/\omega_{min}$will be
- $\frac{1-2C_f}{1+2C_f}$
- $\frac{1-2C_f}{1+2C_f}$
- $\frac{1-C_f}{1+C_f}$
- $\frac{2+C_f}{2-C_fr}$
Multiplication of matrices E and F is G. Matrices E and G are
$E=\begin{bmatrix}
\ cos\theta & -sin\theta & 0 \
\ sin\theta & cos\theta & 0 \
\ 0 & 0 & 1 \
\end{bmatrix}and\hspace{0.1cm} G=\begin{bmatrix}
\ 1 & 0 & 0 \
\ 0 & 1 & 0 \
\ 0 & 0 & 1 \
\end{bmatrix}.\text{What is the matrix F}?$
- $\begin{bmatrix} \ cos\theta & -sin\theta & 0 \\ \ sin\theta & cos\theta & 0 \\ \ 0 & 0 & 1 \\ \end{bmatrix}$
- $\begin{bmatrix} \ sin\theta & cos\theta & 0 \\ \ -cos\theta & sin\theta & 0 \\ \ 0 & 0 & 1 \\ \end{bmatrix}$
- $\begin{bmatrix} \ cos\theta & sin\theta & 0 \\ \ -sin\theta & cos\theta & 0 \\ \ 0 & 0 & 1 \\ \end{bmatrix}$
- $\begin{bmatrix} \ sin\theta & -cos\theta & 0 \\ \ cos\theta & sin\theta & 0 \\ \ 0 & 0 & 1 \\ \end{bmatrix}$
If f (x) = $\frac{2x^2-7x+3}{5x^-12x-9}$, then $\lim \limits_{x\rightarrow3}$f (x) will be
- $\frac{-1}{3}$
- $\frac{5}{18}$
- 0
- $\frac{2}{5}$
Assuming i = $\sqrt{-1}$ and t is a real number, $\int\limits_0^\frac{x}{3}e^{it}dt$is
- $\frac{\sqrt{3}}{2}+i\frac{1}{2}$
- $\frac{\sqrt{3}}{2}-i\frac{1}{2}$
- $\frac{1}{2}+i\frac{\sqrt{3}}{2}$
- $\frac{1}{2}+i\bigg(1-\frac{\sqrt{3}}{2}\bigg)$
In a two-dimensional velocity field with velocities u and n along the x and y directions respectively, the convective acceleration along the x-direction is given by
- u$\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}$
- $U\frac{\partial U}{\partial X}+v\frac{\partial v}{\partial y}$
- u$\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}$
- v$\frac{\partial u}{\partial x}+u\frac{\partial u}{\partial y}$
For a circular shaft of diameter d subjected to torque T, the maximum value of the shear stress is
- $\frac{64T}{\pi d^3}$
- $\frac{32T}{\pi d^3}$
- $\frac{16T}{\pi d^3}$
- $\frac{8T}{\pi d^3}$
A box contains 20 defective items and 80 non-defective items. If two items are selected at random without replacement, what will be the probability that both items are defective?
- $\frac{1}{5}$
- $\frac{1}{25}$
- $\frac{20}{99}$
- $\frac{19}{495}$
A planetary gear train has four gears and one carrier. Angular velocities of the gears are $\infty_1$,$\infty_4$,$\infty_4$ and$\infty_4$ respectively. The carrier rotates with angular velocity $\infty_4$.
What is the relation between the angular velocities of Gear 1 and Gear 4?
- $\begin{matrix} \ \omega_1 -\omega_5 \\ \ \omega_1 -\omega_5 \\ \end{matrix}=6$
- $\frac{\omega_1-\omega_5}{\omega_1\hspace{0.3cm}\omega_5}=6$
- $\frac{\omega_1-\omega_5}{\omega_4-\omega_5}=-\bigg(\frac{2}{3}\bigg)$
- $\frac{\omega_1-\omega_5}{\omega_4-\omega_5}=\frac{8}{9}$
